// Numbas version: exam_results_page_options
{"name": "Divide Polynomials: Cubic by linear", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"advice": "<p>There is a procedure called polynomial long division, if you are comfortable with long division of numbers then it isn't too different. You can see an example of the procedure&nbsp;<a href=\"https://www.khanacademy.org/math/algebra2/arithmetic-with-polynomials/long-division-of-polynomials/v/dividing-polynomials-1\">here</a>. However, if you don't recall long division we&nbsp;can rewrite the numerator so that it includes a multiple of the denominator. We do this term by term starting with the leading term (that is, $\\simplify{{r * m}x ^ 3}$).</p>\n<p>\\[\\begin{eqnarray*} \\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}&amp;=&amp;\\simplify[std]{(x+{s})x^2+{n}x^2+{n*s+t}x+{t*n+be}}\\\\&amp;=&amp;\\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+{t}x+{t*n+be}}\\\\ &amp;=&amp;\\simplify[std]{(x+{s})x^2+(x+{s})*{n}x+(x+{s})*{t}+{t*n+be-s*t}}\\\\ &amp;=&amp;\\simplify[std]{(x+{s})(x^2+{n}x+{t})+{t*n+be-s*t}} \\end{eqnarray*} \\]</p>\n<p>Hence<br/>\\[\\frac{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\\simplify[std]{{r}x+{s}}}=\\simplify[std]{x^2+{n}x+{t}+{t*n+be-s*t}/({r}x+{s})}\\]</p>", "rulesets": {"std": ["all", "fractionNumbers", "!collectNumbers", "!noLeadingMinus"]}, "variables": {"n": {"name": "n", "definition": "random(-9..9)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "al": {"name": "al", "definition": "random(-9..9)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "t": {"name": "t", "definition": "s2*random(-9..9)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "r": {"name": "r", "definition": "1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s1": {"name": "s1", "definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "m": {"name": "m", "definition": "1", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s2": {"name": "s2", "definition": "random(1,-1)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "s": {"name": "s", "definition": "s1*random(1..9)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}, "be": {"name": "be", "definition": "random(-9..9)", "group": "Ungrouped variables", "templateType": "anything", "description": ""}}, "name": "Divide Polynomials: Cubic by linear", "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>Dividing a cubic polynomial by a linear polynomial. Find quotient and remainder.</p>"}, "parts": [{"variableReplacements": [], "marks": 0, "scripts": {}, "showFeedbackIcon": true, "type": "gapfill", "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "gaps": [{"variableReplacements": [], "checkvariablenames": false, "vsetrange": [0, 1], "showFeedbackIcon": true, "marks": 1, "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "showpreview": true, "notallowed": {"showStrings": false, "strings": ["."], "message": "<p>Input numbers as integers not decimals.</p>", "partialCredit": 0}, "type": "jme", "expectedvariablenames": [], "scripts": {}, "vsetrangepoints": 5, "checkingtype": "absdiff", "answersimplification": "std", "checkingaccuracy": 0.001, "answer": "(({m} * (x ^ 2)) + ({n} * x) + {t})"}, {"variableReplacements": [], "mustBeReduced": false, "showFeedbackIcon": true, "marks": 2, "variableReplacementStrategy": "originalfirst", "allowFractions": false, "showCorrectAnswer": true, "mustBeReducedPC": 0, "notationStyles": ["plain", "en", "si-en"], "scripts": {}, "correctAnswerStyle": "plain", "type": "numberentry", "minValue": "{t*n+be-t*s}", "correctAnswerFraction": false, "maxValue": "{t*n+be-t*s}"}], "prompt": "\n      \n      \n      <p>$q(x)=\\;\\;$[[0]]</p>\n      \n      \n      \n      <p>Input all numbers as integers  and not as decimals.</p>\n      \n      \n      \n      <p>$r=\\;\\;$[[1]]</p>\n      \n      \n      "}], "variable_groups": [], "preamble": {"js": "", "css": ""}, "extensions": [], "tags": ["Algebra", "algebra", "algebraic manipulation", "dividing polynomials", "division of polynomials", "polynomial division", "quotient polynomial", "remainder polynomial"], "functions": {}, "ungrouped_variables": ["be", "s2", "s1", "m", "al", "n", "s", "r", "t"], "statement": "<p>Divide $\\displaystyle{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}$ by $\\simplify[std]{{r}x+{s}}$ so that:<br/>\\[\\frac{\\simplify[std]{{r * m} * x ^ 3 + {n * r + m * s} * x ^ 2 + {n * s + t * r} * x + {t * n + be}}}{\\simplify[std]{{r}x+{s}}}=q(x)+\\frac{r}{\\simplify[std]{{r}x+{s}}}\\]</p>\n<p>where $q(x)$ is the quotient polynomial and $r$ is the remainder ($r$ is a constant).</p>\n<p>The coefficients of $q(x)$ are integers, do not input as decimals.</p>", "variablesTest": {"condition": "", "maxRuns": 100}, "type": "question", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}